Math 330:
Exam 1
BIDDLE
Summer 2024
Directions: Hi folks! You can start the exam at 10 am and it should
be completed by 11:50. Everything is graded based on partial credit - if you
do not show work you won’t get credit. No electronic devices are to be used
in completing this examination other than viewing and scanning this exam to
google classroom. Academic dishonesty will be taken very seriously. After I
grade this exam I reserve the right to ask to meet with you for a few minutes
to ask a question or two at random, so please respect academic honesty
policies. Best wishes!!
Problem 1 [16 pts]: Let S be the following relation on the set Z: aSb
iff 5|(a + b)
Determine if S satisfies the indicated property making sure to support
your reasoning:
1. Reflexive
2. Transitive
3. Symmetric
4. Antisymmetric
Problem 2 [16 pts]: Determine which of the following are true and
while are false. If the statement is true give supporting reasons. If the
statement is false give a counterexample or supporting reasons:
1. ∀c, d ∈ N, if c|d then |c| ≤ |d|.
2. If f : X → Y, g : Y → X with f ◦ g = idY then f is onto.
3. The function h : Z → Z given by h(n) = 3n + 1 has a right inverse.
4. Let X, Y, Z be sets. Then
X ∩ (Y ∪ Z) = (X ∩ Y ) ∪ Z.
1
Problem 3 [12 pts]: Given simple statements p, q, r, show that the
following compound statements are logically equivalent:
(¬p ∨ q) =⇒ (p ∧ r)
and
p ∧ (¬q ∨ r)
without using a truth table. If you can’t do this you can construct a truth
table for half the points.
1
Problem 4: Let h : [2, 20] → R be given by the rule h(x) = x−1
4a [9 pts]: Determine if h has a left inverse making sure to support
your reasoning. Determine if h has a right inverse making sure to support
your reasoning.
4b [15 pts]: Determine each of the following, If the quantity cannot be
determined give supporting reasoning:
1. h−1 ([0, 1])
2. h−1 ([2, 4])
3. h([2, 20])
4. h(h−1 ([2, 20])
5. h−1 (∅)
2
Problem 5 [12 pts]: Let R be the following relation on Y = {1, 2, 3, 4, 5}:
1
2
3
4
1
2
3
4
R
Find R−1 and R ◦ R and determine if R is symmetric, antisymmetric,
and/or transitive.
Problem 6 [14 pts]: Let f : X → Y and let C, D ⊆ Y . Prove that
f −1 (C ∩ D) ⊆ f −1 (C).
.
Problem 7 [6 pts]: Let g : R → R be given by g(x) = sin(x). Define
an equivalence relation on R by
x ∼ y ⇐⇒ g(x) = g(y).
].
Find 3 distinct members of each of the equivalence classes [0] and [ 3π
2
3